Download Understanding Hypothesis Testing: Null Hypothesis, Alternate Hypothesis, Test Statistic and more Study notes Mathematical Statistics in PDF only on Docsity! Stat 312: Lecture 10 Hypothesis testing Moo K. Chung mchung@stat.wisc.edu Feb 20, 2003 Concepts 1. The null hypothesis H0 is a claim about the value of a population parameter. The alternate hypothesis H1 is a claim opposite to H0. 2. A test of hypothesis is a method for using sam- ple data to decide whether to reject H0. H0 will be assumed to be true until the sample evidence suggest otherwise. 3. A test statistic is a function of the sample data on which the decision is to be based. 4. A rejection region is the set of all values of a test statistic for which H0 is rejected. 5. Type I error: you reject H0 when H0 is true. P (Type I error) = P (reject H0|H0 true) = α. The resulting α is called the significance level of the test and the corresponding test is called a level α test. We will use test procedures that give α less than a specified level (0.05 or 0.01). In-class problem I believe that dogs are as smart as people. Assume IQ of a dog follows Xi ∼ N(µ, 10 2). IQ of 10 dogs are measured: 30, 25, 70, 110, 40, 80, 50, 60, 100, 60. We want to test if dogs are as smart as people by testing H0 : µ = 100 vs. H1 : µ < 100. One reasonable thing one may try is to see how high the sample mean is. > x<-c(30, 25, 70, 110, 40, 80, 50, 60, 100, 60) > mean(x) [1] 62.5 Since the average IQ of 10 dogs are lower than 100, one would be inclined to reject H0. Let X̄ be a test statistic and R = (−∞, 90] to be a rejection region. Let’s compute the probabil- ity of making Type I error based on this testing procedure. Under the assumption H0 is true, Xi ∼ N(100, 10 2). Under this condition, X̄ ∼ N(100, 10) and α = P (X̄ ≤ 90). > pnorm(90,100,sqrt(10)) [1] 0.0007827011 By using this test procedure, it is highly unlikely to make Type I error. Let’s see what happens when we change the rejection region. When R = (−∞, 95], α = P (X̄ ≤ 95). > pnorm(95,100,sqrt(10)) [1] 0.05692315 When R = (−∞, 99], α = P (X̄ ≤ 99). > pnorm(99,100,sqrt(10)) [1] 0.3759148 The test procedure based on rejecting H0 if X̄ ≤ 99 will produce huge Type I error. Why? Self-study problems Example 8.1., 8.2., 8.3., 8.4., 8.5. Do not compute β.
